Economic
and Game Theory
What is Game Theory?

"You treat world history as a
mathematician does mathematics, in which nothing but laws and formulae
exist, no reality, no good and evil, no time, no yesterday, no
tomorrow, nothing but an eternal shallow, mathematical present." Hermann Hesse

What economists call game theory psychologists call the theory
of social situations, which is an accurate description of what game
theory is about. Although game theory is relevant to parlor games such
as poker or bridge, most research in game theory focuses on how groups
of people interact. There are two main branches of game theory:
cooperative and noncooperative game theory. Noncooperative game theory
deals largely with how intelligent individuals interact with one
another in an effort to achieve their own goals. That is the branch of
game theory I will discuss here.

Decision
theory can be viewed as a theory of one person games,
or a game of a single player against nature. The focus is on
preferences and the formation of beliefs. The most widely used form of
decision theory argues that preferences among risky alternatives can be
described by the maximization of the expected value of a numerical utility
function, where utility may depend on a number of things, but in
situations of interest to economists often depends on money income.
Probability theory is heavily used in order to represent the
uncertainty of outcomes, and Bayes Law is frequently used to model the
way in which new information is used to revise beliefs. Decision theory
is often used in the form of decision analysis, which shows how best to
acquire information before making a decision.

General
equilibrium theory can be viewed as a specialized
branch of game theory that deals with trade and production, and
typically with a relatively large number of individual consumers and
producers. It is widely used in the macroeconomic analysis of broad
based economic policies such as monetary or tax policy, in finance to
analyze stock markets, to study interest and exchange rates and other
prices. In recent years, political economy has emerged as a combination
of general equilibrium theory and game theory in which the private
sector of the economy is modeled by general equilibrium theory, while
voting behavior and the incentive of governments is analyzed using game
theory. Issues studied include tax policy, trade policy, and the role
of international trade agreements such as the European Union.

Mechanism
design theory differs from game theory in that game
theory takes the rules of the game as given, while mechanism design
theory asks about the consequences of different types of rules.
Naturally this relies heavily on game theory. Questions addressed by
mechanism design theory include the design of compensation and wage
agreements that effectively spread risk while maintaining incentives,
and the design of auctions to maximize revenue, or achieve other goals.

One way to describe a game is by listing the players (or
individuals) participating in the game, and for each player, listing
the alternative choices (called actions or strategies) available to
that player. In the case of a two-player game, the actions of the first
player form the rows, and the actions of the second player the columns,
of a matrix. The entries in the matrix are two numbers representing the
utility or payoff to the first and second player respectively. A very
famous game is the Prisoner's Dilemma game. In this game the two
players are partners in a crime who have been captured by the police.
Each suspect is placed in a separate cell, and offered the opportunity
to confess to the crime. The game can be represented by the following
matrix of payoffs

not confess

confess

not confess

5,5

-4,10

confess

10,-4

1,1

Note that higher numbers are better (more utility). If neither
suspect confesses, they go free, and split the proceeds of their crime
which we represent by 5 units of utility for each suspect. However, if
one prisoner confesses and the other does not, the prisoner who
confesses testifies against the other in exchange for going free and
gets the entire 10 units of utility, while the prisoner who did not
confess goes to prison and which results in the low utility of -4. If both prisoners confess,
then both are given a reduced term, but both are convicted, which we
represent by giving each 1 unit of utility: better than having the
other prisoner confess, but not so good as going free.

This game has fascinated game theorists for a variety of
reasons. First, it is a simple representation of a variety of important
situations. For example, instead of confess/not confess we could label
the strategies "contribute to the common good" or "behave selfishly."
This captures a variety of situations economists describe as public
goods problems. An example is the construction of a bridge. It is best
for everyone if the bridge is built, but best for each individual if
someone else builds the bridge. This is sometimes refered to in
economics as an externality. Similarly this game could describe the
alternative of two firms competing in the same market, and instead of
confess/not confess we could label the strategies "set a high price"
and "set a low price." Naturally it is best for both firms if they both
set high prices, but best for each individual firm to set a low price
while the opposition sets a high price.

A second feature of this game, is that it is self-evident how
an intelligent individual should behave. No matter what a suspect
believes his partner is going to do, it is always best to confess. If
the partner in the other cell is not confessing, it is possible to get
10 instead of 5. If the partner in the other cell is confessing, it is
possible to get 1 instead of -4. Yet the pursuit of individually
sensible behavior results in each player getting only 1 unit of
utility, much less than the 5 units each that they would get if neither
confessed. This conflict between the pursuit of individual goals and
the common good is at the heart of many game theoretic problems.

A third feature of this game is that it changes in a very
significant way if the game is repeated, or if the players will
interact with each other again in the future. Suppose for example that
after this game is over, and the suspects either are freed or are
released from jail they will commit another crime and the game will be
played again. In this case in the first period the suspects may reason
that they should not confess because if they do not their partner will
not confess in the second game. Strictly speaking, this conclusion is
not valid, since in the second game both suspects will confess no
matter what happened in the first game. However, repetition opens up
the possibility of being rewarded or punished in the future for current
behavior, and game theorists have provided a number of theories to
explain the obvious intuition that if the game is repeated often
enough, the suspects ought to cooperate.

Some
of the power and meaning of game theory can be illustrated by assessing
the statement "If we were all better people the world would be a better
place." This may seem to you to be self-evidentally true. Or you may
recognize that as a matter of logic this involves the fallacy of
composition: just because a statement applies to each individual person
it need not apply to the group. Game theory can give precise meaning to
the statement of both what it means to be better people and what it
means for the world to be a better place, and so makes it possible to
prove or disprove the statement. In fact the statement is false, and
this can be shown by a variation of the Prisoner's Dilemma.

Let us start with a variation on the Prisoner's Dilemma game we may call the Pride Game.

proud

not confess

confess

proud

4.0, 4.0

5.4, 3.6

1.2, 0.0

not confess

3.6, 5.4

5.0, 5.0

-4.0, 10.0

confess

0.0, 1.2

10.0, -4.0

1.0, 1.0

The Pride Game is like the Prisoner's Dilemma game with the addition of the new
strategy of being proud. A proud individual is one who will not confess
except in retaliation against a rat-like opponent who confesses. In
other words, if I stand proud and you confess, I get 1.2, because we have
both confessed and I can stand proud before your humiliation, but you get 0, because you stand humiliated
before my pride. On the other hand, if we are both proud, then neither
of us will confess, however, our pride comes at a cost, as we both try
to humiliate the other, so we each get 4, rather than the higher value
of 5 we would get if we simply chose not to confess. It would be worse,
of course, for me to lose face before your pride by choosing not to
confess. In this case, I would get 3.6 instead of 4, and you, proud in
the face of my humiliation would get 5.4.

The Pride Game is very different than the Prisoner's
Dilemma game. Suppose that we are both proud. In the face of your
pride, if I simply chose not to confess I would lose face, and my
utility would decline from 4 to 3.6. To confess would be even worse as
you would retaliate by confessing, and I would be humiliated as well,
winding up with 0. In other words, if we are both proud, and we each
believe the other is proud, then we are each making the correct choice.
Morever, as we are both correct, anything either of us learns will
simply confirm our already correct beliefs. This type of situation -
where players play the best they can given their beliefs, and they have
learned all there is to learn about their opponents' play is called by
game theorists a Nash Equilibrium.

Notice
that the original equilibrium of the Prisoner's Dilemma confess-confess
is not an equilibrium of the Pride game: if I think you are going
to confess, I would prefer to stand proud and humiliate you rather than
simply confessing myself.

Now
suppose that we become "better people." To give this precise meaning
take this to mean that we care more about each other, that is, we are
more altruistic, more generous. Specifically, let us imagine that
because I am more generous and care more about you, I place a value
both on the utility I receive in the "selfish" game described above and
on the utility received by you. Not being completely altruistic, I
place twice as much weight on my own utility as I do on yours. So, for
example, if in the original game I get 3 units of utility, and you get
6 units of utility, then in the new game in which I am an altruist, I
get a weighted average of my utility and your utility. I get 2/3 of the
3 units of utility that belonged to me in the original "selfish" game,
and 1/3 of the 6 units of utility that belonged to you in the "selfish"
game. Overall I get 4 units of utility instead of 3. Because I have
become a better more generous person, I am happy that you are getting 6
units of utility, and so this raises my own utility from the selfish
level of 3 to the higher level of 4. The new game with altruistic
players is described by taking a weighted average of each player's
utility with that of his opponent, placing 2/3 weight on his own
utility and 1/3 weight on his opponent's. This gives the payoff matrix of the Altruistic Pride Game

proud

not confess

confess

proud

4.00, 4.00

4.8, 4.20*

0.80, 0.40

not confess

4.20*, 4.80

5.00, 5.00

0.67, 5.33*

confess

0.40, 0.80

5.33*, 0.67

1.00*, 1.00*

What
happens? If you are proud, I should choose not to confess: if I were to
be proud I get a utility of 4, while if I choose not to confess I get
4.2, and of course if I do confess I get only 0.4. Looking at the
original game, it would be better for society at large if when you are
proud I were to choose not to confess. This avoids the confrontation of
two proud people, although of course, at my expense. However, as an
altruist, I recognize that the cost to me is small (I lose only
0.4 units of utility) while the benefit to you is great (you gain 1.4
units of utility), and so I prefer to "not confess." This is shown in
the payoff matrix by placing an asterisk next to the payoff 4.2 in the
proud column.

What should I do if you choose not to confess? If
I
am proud, I get 4.8, if I choose not to confess I get 5, but if I
confess, I get 5.33. So I should confess. Again, this is marked with an
asterisk. Finally, if you confess, then I no longer wish to stand
proud, recognizing that gaining 0.2 by humiliating you comes at a cost
of 1 to you. If I choose not to confess I get only 0.67. So it
is best for me to
confess as well.

What do we conclude? It is no longer an
equilibrium for us both to be proud. Each of us in the face of the
other's pride would wish to switch to not confessing. Of course it is
also not an equilibrium for us both to choose not to confess: each of
us would wish to switch to confessing. The only equilibrium is the box
marked with two asterisks where we are both playing the best we can
given the other player's play: it is where we both choose to confess. So
far from making us better off, when we both become more altruist and
more caring about one another, instead of both getting a relatively
high utility of 4, the equilibrium is disrupted, and we wind up in a
situation in which we both get a utility of only 1. Notice how we can
give a precise meaning to the "world being a better place." If we both
receive a utility of 1 rather than both receiving a utility of 4, the
world is clearly a worse place.

The key to game theory and to
understanding why better people may make the world a worse place is to
understand the delicate balance of equilibrium. It is true that if we
simply become more caring and nothing else happens the world will at
least be no worse. However: if we become more caring we will wish to
change how we behave. As this example shows, when we both try to do
this at the same time, the end result may make us all worse off.

To
put this in the context of day-to-day life: if we were all more
altruistic we would choose to forgive and forget more criminal
behavior. The behavior of criminals has a complication. More altruistic
criminals would choose to commit fewer crimes. However, as crime is not
punished so severely, they would be inclined to commit more crimes. If
in the balance more crimes are committed, the world could certainly be
a worse place. The example shows how this might work.

For
those of you who are interested in or already know more advanced game
theory, the Pride Game has only the one Nash equilibrium shown - it is
solvable by iterated strict dominance. The Atruistic Pride Game, however, has
several mixed strategy equilibria. You can compute them using the fine
open source software program Gambit
written by Richard McKelvey, Andrew McLennan and Theodore Turocy. One
equilibrium involves randomizing between proud and confess, so is worse
than the proud-proud equilibrium of the Pride game. The other is
strictly mixed in that it randomizes between all three strategies. The
payoffs to that equilibrium gives each player 2.31 - so while it is
better than both players confessing for certain, it is still less good
than the unique equilibrium of the Pride Game.

I'd
like to thank Jie Zheng for his help. The "We are all better people"
example is inspired by and based on an academic paper by Sung-Ha Hwang
and Samuel Bowles called "Is Altruism Bad For Cooperation?" If you know
some basic calculus the paper is very readable, and I will provide a
link as soon as they post it online. They provide a much more
persuasive
and robust example of how altruism hurts cooperation, tightly linked to
experimental evidence. If you wish to learn more about game theory, there a variety
of good books on the topic.